The Geometry of Spacetime: A Mathematical Introduction to Relativity Theory

Preface
Introduction
Contents
1 Differentiable Manifolds
	1.1 Charts and Atlases
	1.2 Topologisation
	1.3 Submanifolds of mathbbRm
2 Tangent Vectors
	2.1 The Tangent Space
	2.2 Generation of Tangent Vectors
	2.3 Vector Fields
	2.4 The Lie Bracket
3 Tensors
	3.1 Introduction
	3.2 Multilinear Forms
	3.3 Components
	3.4 Operations with Tensors
	3.5 Tensors on Euclidean Spaces
4 Semi-Riemannian Manifolds
	4.1 Tensor Fields
	4.2 Riemannian Manifolds
	4.3 Bilinear Forms
	4.4 Orientation
	4.5 Spacetime
5 Theory of Special Relativity
	5.1 Kinematics
	5.2 Dynamics
	5.3 Electrodynamics
6 Differential Forms
	6.1 p-forms
	6.2 The Wedge Product
	6.3 The Hodge-Star Operator
	6.4 Outer Derivative
	6.5 The Maxwell Equations in Vacuum
7 The Covariant Differentiation  of  Vector Fields
	7.1 The Directional Derivative in mathbbRn
	7.2 The Levi-Civita Connection
	7.3 Christoffel Symbols
	7.4 The Covariant Derivative on Hypersurfaces
	7.5 The Covariant Derivative in the Schwarzschild Spacetime
8 Curvature
	8.1 The Curvature Tensor
	8.2 The Weingarten Map
	8.3 The Ricci Tensor
	8.4 The Curvature of the Schwarzschild Spacetime
	8.5 Connection Forms and Curvature Forms
9 Matter
	9.1 Mass
	9.2 Energy and Momentum of a Flow
	9.3 The Energy-Momentum Tensor
	9.4 Charge
	9.5 Energy and Momentum in the Electromagnetic Field
	9.6 The Einstein Field Equation
	9.7 Spherically Symmetric Solutions
	9.8 Outer and Inner Schwarzschild Metric
10 Geodesics
	10.1 Time
	10.2 The Euler-Lagrange Equation
	10.3 The Geodesic Equation
	10.4 The Geodesic Deviation
	10.5 Perihelion Precession
	10.6 Light Deflection
	10.7 Red Shift
11 Covariant Differentiation of Tensor Fields
	11.1 Parallel Transport of Vectors
	11.2 Parallel Transport of Tensors
	11.3 Calculation Rules and Component Representation
	11.4 The Second Bianchi Identity
	11.5 Divergence
12 The Lie Derivative
	12.1 The Flow and Its Tangents
	12.2 Pull-Back and Push-Forward
	12.3 Axiomatic Set Up
	12.4 The Derivative Formula
	12.5 Component Representation
	12.6 Killing Vectors
	12.7 The Lie Derivative of Differential Forms
13 Integration on Manifolds
	13.1 Introduction
	13.2 Partition of Unity
	13.3 Integrals
	13.4 Manifolds with Boundary
	13.5 Integral Theorems
	13.6 Extremal Principles
14 Nonrotating Black Holes
	14.1 The Schwarzschild Half Plane
	14.2 Optics of Black Holes
	14.3 The Kruskal Plane
15 Cosmology
	15.1 Spaces of Constant Curvature
	15.2 The Robertson-Walker Metric
	15.3 Universe Models
16 Rotating Black Holes
	16.1 The Kerr Metric
	16.2 Other Representations of the Kerr Metric
	16.3 Causal Structure
	16.4 Covariant Derivative and Curvature
	16.5 Conservation Theorems
17 A Glimpse of String Theory
	17.1 Quantum Theory Versus Relativity Theory
	17.2 Elementary Particles as Strings
	17.3 The Extremal Principle
References
Index